Linear

linear map

Let $V,W$ be vector space over the same field $K$.

A fucntion $f:V\to W$ is said to be linear map if for any two vectors $u,v\in V$ and any saclar $c\in K$ the following two conditions are satisfied:

Additivity- operation of addition$f(u+v)=f(u)+f(v)$
Honogeneity of degree 1 一阶齐次- operation of scalar multiplication,$f(cu)=cf(u)$

linear map preseres保留 the operation of addtion and scalar multiplication

View $K$ as one-dimensional vector space → Linear function

linear extensions

Example: $(1,0)\to -1,(0,1)\to 2,S:=\{(1,0),(0,1)\}$, $F=span(S):(x,y)=x(1,0)+y(0,1)$

A linear extension of $f$ to $X$, if it exists, is a linear map $F:X\to Y$ defined on $X$ that extends $f$ and takes its values from the codomain {上域$X$} of $f$

When the subset $S$ is a vector subspace of $X$ then a linear extension of $f$ to all of $X$ is genranted to exist iff $f:S\to Y$ is a linear map.
The map$f:S\to Y$ can be extended to a linear map $F:span S\to Y$ iff $\forall n>0,c_i \text{ are scales}, s_i\in S ,\text{such that }0=\sum c_is_i\text{ and necessarily}\{0=\sum c_if(s_i)\}$
if linear extension of $f:S\to Y$ exists, then $F:span S\to Y$ is unique and

$$ F\left(c_1s_1+\cdots c_ns_n\right)=c_1f\left(s_1\right)+\cdots+c_nf\left(s_n\right) $$

Matrix representaion

if $V,W$ are finite-dimensional vector spaces and basis is defind for each vector, then every linear map from $V$ to $W$ can be represented by a matrix.

Transformation Matrix

Vector space of linear map

the composition of lineat maps is linear: $f:V\to W,g:W\to Z$ are linear ⇒$g\circ f:V\to Z$ is linear